where \(\widetilde{V}_{\mathrm {c},\kappa }= e^{i2\pi \kappa Q_{\mathrm {c}}\otimes Q_{\mathrm {c}}}\)  as in Section .
For the proof, see [1]}. The proof is written again for
an action of \(S^1\) , however, the adaptation is easy for \({\mathbb {Z}}_N\) . Note that one has to consider
spectral subspaces of \(V_{\mathrm {c}}\)  parametrized by \({\mathbb {Z}}_N\) . The identification
\({\mathbb {Z}}_N = {\mathbb {Z}}/ N{\mathbb {Z}}\)  should be always kept in mind and numbers, e.g., \(e^{i\frac{2\pi k}{N}}\) 
are well-defined for \(k\in {\mathbb {Z}}_N\) .
